Discrete Applied Mathematics 48 (1994) 2455259

North-Holland

245

A set covering reformulation of the pure fixed charge transportation problem

Maud Giithe-Lundgren and Torbjijrn Larson

Department of Mathematics, Linkiiping Institute of Technology. S-581 83 Linksping. Sweden

Received 15 September 1989

Revised 1 December 1990

Abstract

The pure fixed charge transportation problem is reformulated into an equivalent set covering

problem with, in general, a large number of constraints. Two constraint generation procedures for its

solution are suggested; in both of these, new constraints are generated by the solution of ordinary

maximum flow problems. In the first procedure, which is an optimizing Benders-type scheme, each

set covering problem is solved by a simple heuristic. Upper and lower bounds to the optimal value of

the transportation problem are provided throughout the procedure so that it can be terminated at

a-optimality. The second procedure, which is a heuristic, is based on the restricted Lagrangean

concept. In this case the generated constraints are Lagrangean relaxed with fixed multiplier values.

These are chosen so that the optimal solution of an initial Lagrangean subproblem, being a min-

imum cost network flow problem, remains optimal. At termination, the heuristic provides a lower

bound and a feasible solution to the transportation problem. Moreover, the computational cost of

this procedure is very low; hence it is well suited for incorporating in a branch and bound scheme.

A possible branching technique is given. Computational results are presented for both the Benders-

type and the branch and bound schemes.

Keywords. Pure fixed charge transportation problem, constraint generation, set covering problem,

Lagrangean duality, restricted Lagrangean.

The set covering reformulation

Consider a transportation network with a set of sources, I, and a set of sinks, J. A supply of Si units is associated with each source i E I and a demand Dj with each

sink j E J. Moreover, when any positive flow is distributed along an arc (i, j), the fixed

COstfij 2 0 arises. This cost is independent of the amount of flow that traverses the arc,

which is said to be open when it is allowed to carry positive flow. The aim is to

determine a set of open arcs which can be used to satisfy the demands at the sinks to

a minimal total cost. This problem is recognized as the pure fixed charge transporta-

tion problem (PFCTP).

0166-218X/94/$07.00 0 1994-Elsevier Science B.V. All rights reserved

SSDI 0166-218X(92)00177-9

246 M. Giithr-Lundgren, T. Larsson

This problem has not been studied in great detail in the past, although a solution

method based on direct search has been suggested by Fisk and McKeown [S].

However, the closely related fixed charge transportation problem in which each arc

has a fixed cost as well as a linear cost has been studied to a greater extent. Exact

solution methods have been presented by Gray [ 111, Kennington and Unger [ 141 and

Murty [17] among others. Early heuristics were given by Balinski [4] and Kuhn and

Baumol [15]. A basic property of all linear fixed charge problems is that the optimum

is attained at an extreme point of the feasible region of the continuous variables [13].

One particular difficulty of fixed charge transportation problems is that degeneracy

regularly occurs. Approaches for handling this phenomenon have been proposed by

Ahrens and Finke [l] and by McKeown [16].

As Fisk and McKeown [S] have mentioned, a good algorithm for the solution of

some special class of fixed charge transportation problems does not have to be

efficient for solving all problem instances. Some algorithms may work better than

others for a given relation in the size of the fixed and linear cost coefficients. This fact

motivates the development of specialized algorithms for the pure fixed charge trans-

portation problem.

The PFCTP can be stated mathematically as

S.t. ~JXij = Si, i E I,

2 xij = Dj, j E J,

Xij 0, i E I, j E J,

yij = 1 if Xij > 0 and 0 otherwise, i E I, j E J,

(PFCTP)

where Cisl Si = CjsJ Dj.

In the following, we will derive a formulation of the PFCTP in the binary variables

only. The feasible set of this pure integer program, denoted Y, is the projection of the

feasible set of PFCTP onto the vertices of the 1 II x 1 J I-dimensional hypercube, i.e.,

Y= {yE{0,1}“‘X’J’~x20 exists, such that 1 Xij = Si, i E I,

D gxij = j> J E J, and Xij = 0 if Yij = 0, i E I, j E J}.

This implicit definition is clearly not appropriate for computational purposes and

we will therefore deduce an equivalent explicit description by using the following

theorem.

Theorem 1. In a capacitated transportation network, with arc capacities kij 2 0, i E I,

j E J, there is a feasible pow consistent with all node supplies and demands if and only if

z min(Si, C kij) 2 C Dj, for al/ L E J. jtL jsL

The pure ,j.ued charge transportation problem 247

The theorem, which is a special case of a result given by Gale [9], is an application

of Farkas’ lemma to a capacitated transportation network. In the pure fixed charge

transportation problem, the capacity of an arc (i,j), i E I, j E J, is infinite if yij = 1 and

zero otherwise. The following theorem states the equivalent representation of the

set Y.

Theorem2. I’= (YE{O,I}‘~‘~‘~‘IC~~~C~~~Y~~~ lfora/lL~ JandKcIsuchthat

Ci,xS’i < CjeLDj}, where IT = I - K.

Proof. Clearly,

Emin/Si, C kij) I 1 Si + x_ 1 kij, VK c 1, jeL isK ieK jeL

with equality for some K, so that the condition of Theorem 1 can be restated as

C C kij 2 C Dj - C Si, VL c J and VK c I. ieK jeL jtL ieK

If for a particular choice of L and K the right-hand side is nonpositive, the condition is

always fulfilled, and otherwise

C 1 Yij 2 l isK jtL

must hold since all capacities are infinite or zero. This proves the theorem. 0

The set covering inequalities of this representation have a very simple interpreta-

tion; if the total demand of a set of sinks, L, is larger than the total supply of a set of

sources, K, then there must be at least one open arc from K to L in any feasible

solution to PFCTP. This type of valid inequalities has also been observed by Padberg

et al. [18], in their study of facets for fixed charge problems.

The theorem leads to the following set covering reformulation of the PFCTP.

min C C fijYij9

isI jsJ

s.t. iT1, j’$ yij 2 1, V L c J and V K z I such that iz Si < ,~L Dj, (SCP)

yij E (0, l}, i E I, j E 1.

The number of constraints in this formulation may of course be very large. The

constraints can, however, be generated algorithmically as shall be demonstrated in the

next section. It should be noted that the above reformulation and the constraint

generation procedure to be presented have great similarities to Benders’ decomposi-

tion principle [6].

248 M. GBthe-Lundgren, T. Larsson

A constraint generation procedure for SCP

In order to initiate the procedure, a relaxed set covering problem with only a subset

of the constraints of those appearing in SCP should be solved. One possible way to

construct this initial relaxed problem is to utilize the fact that any feasible solution to

PFCTP must satisfy the following knapsack constraints.

1 Djyij 2 Si, i E I, js.l

ZSiyij 2 Dj, jE J.

For each knapsack constraint we construct a minimal cover [2]. For a knapsack

constraint with index i, a cover Ji G J is minimal if ‘&sj, Dj < Si and &ej, Dj + D,

2 Si for any q E Ji, where Ji is the complement of Ji. A cover Zj G I for a knapsack

constraint j is constructed analogously. The minimal covers are used for defining the

initial relaxed set covering problem which has the following form.

s.t. jz _Yij 2 1, i E I,

iz.Yij 2 1, je J, (MCNFP)

J

Yij E (0, l}, i E I, j E J.

Here, Ji c J for all i E I, and Ij G I for all j E J. This initial set covering problem can

be solved as a minimum cost network flow problem since each variable appears once

at most in each set of constraints. It is a relaxation of SCP and hence will produce

a lower bound to the original problem.

Now assume that any relaxed set covering problem has been solved, giving an

optimal solution y* and a corresponding lower bound. The problem of checking

whether a feasible flow corresponding to y* exists or not, can be solved as a maximum

flow problem. The technique used resembles the solution of the restricted primal

problem in the primal-dual algorithm when applied to an ordinary transportation

problem (see e.g. [19]). The maximum flow problem can be stated as

max c xij3

(i. j)tIJ*

S.t. ,FTXij I Si, i E 1,

zxij 2 Dj, .iE J,

xij 2 0, (i, j)e IJ*,

xij= 0, (i,j)EIJ*,

WFP)

The purejkd charge transportation problem 249

where ZJ* = {( i,j) 1 yfj = 1) and IJ* its complement. It is solved in the partial graph

induced by y*, augmented by adding a supersource s and a supersink t. The arcs (s, i),

i E I, will have capacities equal to Si and the arcs (j, t), j E J, have capacities equal to

Dj. When MFP is solved, giving a flow x*, either of the following two cases will occur.

The proofs are straightforward.

Theorem 3. If the maximumJlow equals c ie~ Si, then y* is feasible in SCP and therefore (x*, y*) is optimal in PFCTP.

Theorem 4. If the maximum flow is strictly less than Lit, Si, then y* is not feasible in

SCP and there exists a minimal cut C(K, L) corresponding to x* giving a violated set

covering inequality.

Thus the separation problem to be solved in each iteration is a standard maximum

flow problem. In the following, we will call this the feasibility test for a solution y*, and

the corresponding valid inequality the feasibility cut. The feasibility cut provided by

the solution of the maximum flow problem can often be strengthened heuristically.

This is done by removing as many sources and sinks from K and L, respectively, as

possible, while maintaining xisK Si < Cj,LDj for the updated sets K and L. When the

sets K and L cannot be further reduced, the resulting feasibility cut is said to be

minimal. Clearly, if the original sets &? and L can be reduced, then the minimal

feasibility cut is stronger than the original one. The algorithm proceeds by the

addition of a minimal feasibility cut and a new set covering problem is solved to

generate a new solution y*. Row reduction tests (see e.g. [lo]) can be applied in order

to reduce the number of constraints each time a new cut has been generated.

Since a set covering problem is to be solved in each iteration, it is not practical to

require that the problem should be solved to optimality. Instead, we make use of one

of the many existing heuristics for set covering problems [7]. Since there is no

guarantee that the solution generated is a true optimal solution to the current set

covering problem, we need to modify the conclusion reached once a feasibility test is

passed. If the heuristic solution is feasible, we only know that it gives an upper bound

to the optimal value of PFCTP. Hence it is stored as a candidate for optimal solution.

(Note that since an arc that is open according to the heuristic set covering solution

might be unused by the maximum flow, this upper bound could actually be lower than

the cost of the heuristic solution.)

Moreover, each time a feasible solution to PFCTP is found, we try to cut it off using

a set covering constraint which is a somewhat improved version of the cutting plane

developed by Bellmore and Ratliff [S]. It is constructed as follows. The set covering

heuristic we have used will always give a nonredundant solution, that is an integer

solution which is a vertex to the feasible polyhedron of the continuous relaxation of

the current set covering problem. It is then possible to construct a so-called involutory

basis for this basic solution and compute the corresponding reduced costs, say xj,

i E I, j E J. The reduced costs will be integral assuming that the original cost coeffi-

250 M. GBthe-Lundgrm, T. Larsson

cients are integral. Further, let zh be the objective value of the heuristic solution and let

Z be the current upper bound. A necessary condition for any set covering solution

being better than the current upper bound is then given by

zh+ 1 CJjYijIZ-1 it1 jeJ

which implies that

- 1 fijyij 2 zh-?+ 1 (i, j)eIJ-

where ZJ- = {(i,j) IJj < O}. A set covering constraint that cuts off the heuristic

solution is then obtained by constructing a minimal cover to this knapsack constraint

[2]. We will in the following refer to this minimal cover inequality as the exclusion cut.

Clearly, if IJ- or the minimal cover is the empty set, then the current upper bound has

been shown to be optimal and the solution procedure is terminated. However, an

exclusion cut will usually be generated. A basic property of this cutting plane is that it

cuts off integer points, although no solution which is better than the heuristic one will

be cut off, since each cutting plane defines a necessary condition for an improved

solution. By use of the exclusion cuts, it is ensured that no heuristic solution is

repeated. The finiteness of this scheme follows from the facts that SCP has a finite

number of constraints and that the Bellmore-Ratliff procedure is finite. Ultimately,

the above-mentioned optimality verification will cause termination.

However, this optimality verification is, from a practical point of view, not so useful

since for large-scale problems it will usually yield a very large number of iterations.

We have therefore used a dual subgradient procedure to work parallel to the

generation of minimal feasibility cuts and exclusion cuts, to provide an alternative

termination criterion. This has turned out to work well in practice. Let P denote the

current set of feasibility and exclusion cuts. These inequalities are written in the

generic form

CCatYij21, PEP. isl jeJ

The optimal multipliers of the Lagrangean dual

maxh(w)=minC CfijYij+ C Wp

WZO isl jsJ PEP

,

S.t. CYij21, iEl, jsJ,

yijE{O,l}, igZ,jcJ

The pure fixed charge transportation problem 251

are calculated by using a subgradient procedure [12]. At iteration t, a subgradient to

the function h(w) is computed as

g!’ = 1 - C 1 a$yiy, p E P, itIjeJ

where y (I) is the solution of the Lagrangean subproblem, corresponding to multiplier

values w(‘). The new values of the multipliers are given by

wF+‘) = max(O, wg) + 0gg)), p E p,

where Q can be computed as

e = P(4wop’) - h(w”‘)) 11 g(l) 11 2 ’ 0 < p < 2.

Since the optimal multipliers wop’ are unknown, the value h( wop’) is approximated by

the objective value of the most recent heuristic solution. The subgradient procedure is

terminated after a fixed number of iterations, which depend on the problem size.

Since the set P includes constraints which cut off feasible solutions, the lower bound

obtained from the Lagrangean dual will in fact at some point of the solution process,

exceed the value of the optimal solution to PFCTP. When this situation occurs, the

procedure can be terminated, since it can be guaranteed that the optimal solution to

any subsequent set covering problem will not be better than the one saved as

a potential optimal solution. Thus, this solution is indeed optimal for the PFCTP. We

will therefore refer to this bound as the termination bound and it can naturally also be

used as a criterion for s-optimality.

In Fig. 1 the constraint generation method for SCP is summarized.

A restricted Lagrangean method

The second method to be presented is based on the restricted Lagrangean principle

[3]. The idea here is to solve a relaxation of SCP which has the integrality property

and then to identify violated set covering constraints. These inequalities are dualized

with multipliers which are constrained so as to guarantee the continued optimality of

the initial solution. Therefore, no repeated solution of the Lagrangean subproblem is

required. This procedure gives a lower bound on the optimal objective function value.

Further, when no more violated inequalities can be generated, it provides a feasible

solution to PFCTP and a corresponding upper bound. The computational cost of the

method is very low. The restricted Lagrangean method can be used as a heuristic or

included in an optimizing branch and bound procedure. For the latter case we give

a possible branching technique.

252

I

M. Gdthe-Lundgren,

A I I

T. Larsson

I Generate a heuristic solution to the current set covering

1 If the termination bound exceeds

Generate a termination bound by the subgradient procedure.

I

Add a minimal feasi- bility cut. Apply the reduction tests.

I ” Feasibility test: Solve a maximum flow problem on the

I no

partial graph. Is the flow feasible?

I yes

Store the best upper bound.

Exclusion procedure: Add an exclusion cut that eliminates the heuristic solution. Apply the reduction tests.

I

A

Fig. 1

The set covering reformulation of the PFCTP can be stated as

min 5 zJ.fli Yij,

s.t. jJ$ Yij 2 l, i l I,

iz Yij 2 l2 jc J, I

C CdjYij2 19 PEP, itl jEJ

Yij E (0, l}, i E I, j E J,

where the third set of constraints is the set covering inequalities of SCP written in

generic form. Let the relaxed problem defined by omitting these constraints, i.e., the

problem MCNFP stated previously, have the optimal solution y*.

The pure fixed charge transportafion problem 253

The Lagrangean dual with respect to the generic constraints can be formulated as

follows.

maxh(w)= C w,+minC C(fij- C WpU$)yij,

W>O PEP isI jtJ PEP

S.t. C Yij 2 1, i E 1,

jeJ,

iz Yij 2 1, j E JT I

Yij E (0, l}, i E I, j E J.

(D)

A solution to this problem gives a lower bound on the optimal objective function

value of PFCTP. The restricted Lagrangean dual problem is given by

maxh(w) wsw

(RD)

where

W = {w E Rip’ ( w 2 0 and y* solves the Lagrangean subproblem}.

The continued optimality of y* can of course be easily stated in terms of dual

feasibility and complementary slackness since the Lagrangean subproblem has the

integrality property. The optimal value of RD gives a lower bound on the optimal

value of D, thus also a lower bound on the optimal value of PFCTP. We shall now

present a constraint generation procedure which approximates the optimum of RD.

The procedure is initialized by solving MCNFP which gives the optimal solution y*

and optimal reduced costs ~j, i E I, j E J. Since fij 2 0 for all i and j, the bounds

yii I 1, i E I, j E J, are in fact redundant so that _@ 2 0 holds. The upper bounds will

remain redundant throughout the procedure. Given the initial solution, we success-

ively identify any minimal feasibility cuts that

(i) are not satisfied by the solution y*,

(ii) admit a positive multiplier wp which together with w, , . . . , wp- 1 satisfies w E W.

Such a cut can be identified by performing the feasibility test on the solution y’ given

by

Yij = ’ i

1, if fi”;- = 0,

0, otherwise.

The minimal cut in the partial graph yields a feasibility cut which is reduced to

a minimal feasibility cut, as described in the previous section. Let C( K, L) be the cut

set corresponding to the minimal feasibility cut. The largest value that can be assigned

to the multiplier of the new minimal feasibility cut p, without violating the continued

optimality of y*, is

wp = min fij > 0. ieK, jtL

254 M. Gbthe-Lundgren, T. Larsson

Compute the reduced costs and generate the solution y'.

Generate a minimal feasibility cut. Fix the value of the new multiplier and add the constraint to the objective function of RD.

Feasibility test: Solve a maximum flow problem on the partial graph induced by y'. Is the flow feasible?

yes

Check the solution to PFCTP for optimality and terminate. I

Fig. 2.

The objective value of RD will then increase by wp. By updating the reduced costs,

fi”;. := fij - w,, for all arcs (i, j) such that i E K, j E L, we obtain a new solution y’ to test

for feasibility. The new maximum flow problem is easily reoptimized from the

previous one. The procedure is repeated until no more feasibility cuts can be found,

and then the lower bound given by RD is stored.

At the point when no more cuts can be identified, the corresponding maximum flow

solution gives a heuristic solution to the PFCTP. This solution can be tested for

optimality; it is an optimal solution if all the inequalities associated with positive

multipliers are satisfied with equality.

The restricted Lagrangean procedure is summarized in Fig. 2.

When the procedure terminates without verifying optimality, a branch and bound

procedure can be applied in order to close the duality gap. A possible branching rule

can be derived from the minimal feasibility cut obtained by performing the feasibility

test on the initial solution y*. The inequality says that at least one of the variables

included must take the value one. Thus, if A is the set of arcs passing the minimal cut

in the forward direction, then every feasible solution to the PFCTP satisfies the

disjunction

VI (Yi(n)j(n) = 1 and Yi(r)j(r) = 0, vr 5 71 - 1).

The pure fixed charge transportation problem 255

This branching technique is analogous to the one derived from a subtour breaking

inequality for the travelling salesman problem. If, at any node of the search tree, some

of the arcs included in A were already fixed at zero, they are removed from set A. This

branching strategy creates at most 1 A I subproblems, and in all of these the solution y*

becomes infeasible. Clearly, the branch and bound procedure can be applied as

a truncated procedure, and thereby an s-optimal solution is obtained.

Computational results

In order to evaluate the two suggested methods, a set of randomly generated

problems of varying sizes, was used. The problems were generated according to

Fisk and McKeown [8]. That is, the fixed charges were uniformly generated with

integer values between 0 and 10 and the demands were uniformly generated with

values between 10 and 100 in increments of 10. The supplies were generated in

a similar manner in such a way that total supplies equalled total demands. 13 sets of

6 problems each were generated such that the number of O/l-variables was between 25

and 400.

The results for the first method are shown in Table 1. Since none of the problems in

the size 200-400 O/l-variables were solved to optimality within the given limit of

iterations, Table 1 includes only results for the group of smaller problems (25-100

O/l-variables). The number of each type of constraint generated is given as an average

of the results for the problems of the same size. “Total” means simply the average

number of constraints generated, and “(min, max)” are the minimum and maximum

numbers of constraints that were generated for any problem of the corresponding size.

We also give averages for the number of constraints eliminated by row reduction tests

and constraints generated before optimum was found. No CPU-times are reported

since the developed code is purely experimental. The maximum number of iterations

was fixed to 200 and for the problems not solved to optimality, we give the average of

the deviation of lower and upper bounds.

In Tables 2 and 3 the results of the restricted Lagrangean approach combined with

branch and bound are shown. The branching technique used was the one presented

above. Branching was always performed from the node which had the smallest lower

bound. We give average numbers of generated constraints, solved subproblems and

nodes in the search trees, as well as average total CPU-times for the solution of the

minimum cost network flow and maximum flow problems, respectively. The solution

of these two problems is the time-consuming part of the algorithm. Both problems

were solved using a primal network simplex code written in Fortran.

All the problems which included 255100 O/l-variables have also been solved using

the ZOOM/XMP-package developed by R. Marsten. The corresponding CPU-

times are reported as an average of the results for the problems of the same size.

Finally, the CPU-times which are available from Fisk and McKeown [S] are also

given.

Tab

le

1

Prob

lem

si

ze

Feas

ibili

ty

Exc

lusi

on

Tot

al

(min

, m

ax)

Red

uced

B

efor

e

optim

um

No.

of

pro

b-

lem

s so

lved

with

in

200

itera

tions

(UB

D

- L

BD

)/L

BD

5x5

5

5x7

7

5 x

10

15

5x

15

31

5 x

20

24

6x6

18

1x1

33

8x8

22

3 8

W6)

3

5

3 10

W

3)

4 6

15

30

(143

40)

7 21

26

57

(9,1

67)

10

55

6 30

(2

9331

) 3

29

25

43

(9,9

9)

12

18

66

99

(19,

179)

34

27

7 29

(l

&37

) 5

13

6 6 _

6 4 10

%

2 6%

6

_

5 10

%

4 6%

Tab

le

2

Prob

lem

si

ze

Con

stra

ints

Su

bpro

blem

s N

odes

T

ime-

MC

NFP

(s)

Tim

e-M

FP

(s)

Tim

e-

ZO

OM

/XM

P

Tim

e-Fi

sk

and

McK

eow

n”

(s)

5x5

1 14

5

0.09

0.

06

2.85

0.

196

5x7

4 14

5

0.11

0.

07

7.09

0.

467

5x10

15

40

12

0.

35

0.30

11

.22

0.37

7 5x

15

49

83

19

0.

84

0.85

33

.34

_

5 x

20

553

491

87

6.16

8.

17

236.

78

_

6x6

7 35

12

0.

30

0.20

38

.84

0.59

7

7x7

22

321

53

3.14

0.

79

687.

85

2.45

1

8x8

21

115

23

1.37

0.

60

494.

40

1.87

3

“A C

YB

ER

70

174.

--

-.

^.

_.

.^

The pure fixed charge transportation problem 257

Table 3

Problem Cons-

size traints

Sub- Nodes Time- Time-MFP No. of problems (UBD - LBD)/

problems MCNFP (s) solved to LBD

(s) optimality

10x20 453 2878 154 64.9 11.8 5 0.06

10x30 2125 7768 383 218.0 59.5 4 0.15

10x40 1746 9511 248 311.1 64.1 2 0.10

15x 15 1625 4985 405 155.0 39.0 5 0.06

20 x 20 1658 7393 393 340.8 55.6 2 0.17

We have not taken advantage of reoptimization possibilities in any part of the

algorithm. The computations were made on a SUN 4/390 and all the codes were

written in Fortran. The search tree is limited to 1000 nodes and for the problems that

are not solved to optimality the average of the deviation of lower and upper bounds is

given.

At each node of the search tree, one single minimum cost network flow problem and

a number of maximum flow problems are solved. For the given test problems, the

number of maximum flow problems which are solved at one node is in the range

between zero and six. In general, the number of maximum flow problems which are

solved decreases when the number of variables, fixed to zero or one, increases.

A general observation is that the LP-bound for those problems which have

approximately the same number of sources and sinks is relatively weak compared to

the LP-bound for problems with more sinks than sources. This observation would

explain the high CPU-times spent on solving the former problems with the

ZOOM/XMP-package. However, the results obtained from the restricted Lagran-

gean procedure do not show any significant difference which is due to the shape of the

test problems.

Conclusions and discussion

A reformulation of the pure fixed charge transportation problem into an equivalent

set covering problem, with a large number of constraints, has been presented. The

PFCTP is usually stated as a mixed integer program, but since there are no costs

associated with the continuous variables of this program, the reformulation into

a pure integer program is, from an intuitive point of view, a quite natural step. Two

constraint generation procedures were given, one Benders-type scheme and one

restricted Lagrangean heuristic. The latter was combined with a branch and bound

scheme. The computational results show that both methods are feasible approaches to

the solution of the PFCTP; however, when used for large-scale problems the first one

does not seem to be efficient.

258 M. Giirhe-Lundgren, T. Larson

When applied to large-scale problems, the Benders-type procedure will always

require a great number of iterations before the first feasible solution to the original

problem is found. In that case it would probably be suitable to add to the procedure

some simple heuristic for converting each heuristic set covering solution into a feasible

solution to the PFCTP. This modified method could then be used as a heuristic

providing a lower and an upper bound. The procedure can be stopped before

optimality is verified and usually a good solution has already been obtained.

The PFCTP is, in fact, a single-commodity uncapacitated network design problem

in a bipartite network. An interesting subject for future research would therefore be to

extend our approaches to more general network design problems, with for instance

multiple commodities, arc capacities or nonbipartite networks.

Acknowledgement

The research leading to this paper

Board for Technical Development (STU

Council (NFR). We are thankful to

suggestions.

was supported by the Swedish National

.) and the Swedish Natural Science Research

the anonymous referees for their helpful

References

[1] J.H. Ahrens and G. Finke, Degeneracy in fixed charge transportation problems, Math. Programming

8 (1975) 369-374.

[2] E. Balas, Facets of the knapsack polytope, Math. Programming 8 (1975) 146-164.

[3] E. Balas and N. Christofides, A restricted Lagrangean approach to the traveling salesman problem,

Math. Programming 21 (1981) 19-46. [4] M.L. Bahnski, Fixed-cost transportation problems, Naval Res. Logist. Quart. 8 (1961) 41-54.

[S] M. Bellmore and H.D. Ratliff, Set covering and involutory bases, Management Sci. 18 (1971)

194-206.

[6] J. Benders, Partitioning procedures for solving mixed-variables programming problems, Numer.

Math. 4 (1962) 2388252. [7] V. Chvital, A greedy heuristic for the set-covering problem, Math. Oper. Res. 4 (1979) 233-235.

[S] J. Fisk and P.G. McKeown, The pure fixed charge transportation problem, Naval Res. Logist. Quart.

26 (1979) 631-641.

[9] D. Gale, A theorem on flows in networks, Pacific J. Math. 7 (1958) 1073-1082.

[lo] R. Garfinkel and CL. Nemhauser, Integer Programming (Wiley, London, 1972).

[11] P. Gray, Exact solution of the fixed-charge transportation problem, Oper. Res. 19 (1971)

152991538. [12] M. Held, P. Wolfe and H.P. Crowder, Validation of subgradient optimization, Math. Programming

6 (1974) 62-88. [13] W.M. Hirsch and G.B. Dantzig, The fixed charge problem, Naval Res. Logist. Quart. 15 (1968)

413-424.

[14] J.L. Kennington and V.E. Unger, A new branch-and-bound algorithm for the fixed-charge trans-

portation problem, Management Sci. 22 (1976) 1116-l 126. [15] H.W. Kuhn and W.J. Baumol, An approximative algorithm for the fixed-charges transportation

problem, Naval Res. Logist. Quart. 9 (1962) l-16.

The pure ,$xed charge transportation problem 259

[16] P.G. McKeown, Some computational results of using the AhrensFinke method for handling

degeneracy in fixed charge transportation problems, Math. Programming 15 (1978) 3555359.

[17] K.G. Murty, Solving the fixed charge problem by ranking the extreme points, Oper. Res. 16 (1968)

268-279. [18] M.W. Padberg, T.J. Van Roy and L.A. Wolsey, Valid linear inequalities for fixed charge problems,

Oper. Res. 33 (1985) 842-861. [19] C.H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity

(Prentice-Hall, Englewood Cliffs, NJ, 1982).